5 Exponent pairs

Definition 5.1 Exponent pair

An exponent pair is a (fixed) element \((k,\ell )\) of the triangle

\begin{equation} \label{exp-pair-triangle} \{ (k,\ell ) \in \mathbf{R}^2: 0 \leq k \leq 1/2 \leq \ell \leq 1, k+\ell \leq 1 \} \end{equation}

with the following property: for all model phase functions \(F\), all \(T \geq N \geq 1\), and all intervals \(I \subset [N,2N]\), one has

\begin{equation} \label{ntf} \sum _{n \in I} e(T F(n/N)) \ll (T/N)^{k+o(1)} N^{\ell +o(1)} \end{equation}

whenever \(T \geq N \geq 1\), \(I\) is an interval in \([N,2N]\), and \(F \in {\mathcal U}\).

Implemented at exponent_pair.py as:
Exp_pair

One can formulate the notion of an exponent pair without recourse to asymptotic notation:

Lemma 5.2 Non-asymptotic definition of exponent pair

Let \((k,\ell )\) be a fixed element of 1. Then the following are equivalent:

\((k,\ell )\) is an exponent pair.
For every (fixed) \(\varepsilon {\gt}0\) there exist (fixed) \(C, P {\gt} 0\) such that, whenever \(T \geq N \geq 1\), \(I \subset [N,2N]\), and \(F\) is a phase function obeying 3 for all (fixed) \(0 \leq p \leq P\) and \(u \in [1,2]\), then

\[ |\sum _{n \in I} e(T F(n/N))| \leq C (T/N)^{k+\varepsilon } N^{\ell +\varepsilon }. \]

The proof of this lemma is similar to that of Lemma 4.3 and is omitted.

Exponent pairs are closely related to the function \(\beta \) from the previous chapter:

Lemma 5.3 Duality between exponent pairs and \(\beta \)

Let \((k,\ell )\) be in the triangle 1. Then the following are equivalent:

\((k,\ell )\) is an exponent pair.
\(\beta (\alpha ) \leq k + (\ell -k)\alpha \) for all \(0 \leq \alpha \leq 1\).

Implemented at exponent_pair.py as:
exponent_pairs_to_beta_bounds()
beta_bounds_to_exponent_pairs()

Thus exponent pairs are dual to the convex hull of the graph of \(\beta \). But \(\beta \) is not known to be convex, so one could have bounds on \(\beta \) that do not directly correspond to exponent pairs. We remark that in the case \(\ell - k \geq 1/2\), one only needs to check the case \(0 \leq \alpha \leq 1/2\) in (ii) above, since the remaining regime \(1/2 \leq \alpha \leq 1\) then follows from Lemma 4.10 and some algebra. Conversely, if \(\ell - k \leq 1/2\), one only needs to check the region \(1/2 \leq \alpha \leq 1\).

Proof ▶

If (i) holds, then for any \(0 {\lt} \alpha {\lt} 1\), any unbounded \(T \geq 1\), any \(N = T^{\alpha +o(1)}\), interval \(I \subset [N,2N]\), and model phase function \(F\), we have from (i) that

\[ \sum _{n \in I} e(T F(n/N)) \ll (T/N)^{k+o(1)} N^{\ell +o(1)} = T^{k + (\ell -k)\alpha + o(1)}. \]

From Definition 4.2 we conclude that \(\beta (\alpha ) \leq k + (\ell -k) \alpha \). Also since \((k,\ell )\) lies in 1, we see from 4, 8 that we also have \(\beta (\alpha ) \leq k + (\ell -k) \alpha \) for \(\alpha =0,1\).

Now suppose that (ii) holds. Let \(F, T, N, I\) be as in Definition 5.1. By underspill it suffices to show that

\[ \sum _{n \in I} e(T F(n/N)) \ll (T/N)^{k+\varepsilon +o(1)} N^{\ell +\varepsilon +o(1)} \]

for any fixed \(\varepsilon {\gt}0\). We may assume that \(T \leq N^{1/\varepsilon +1}\), since the claim follows from the trivial bound \(\sum _{n \in I} e(T F(n/N)) \ll N\) otherwise. We may also assume that \(N\) is unbounded, since the claim is clear for \(N\) bounded; this forces \(T\) to be unbounded as well.

By passing to a subsequence we may assume that \(N = T^{\alpha +o(1)}\) for some fixed \(0 \leq \alpha \leq 1\). By Definition 4.2 we then have

\[ \sum _{n \in I} e(T F(n/N)) \ll T^{\beta (\alpha )+o(1)} \]

and hence by (ii)

\[ \sum _{n \in I} e(T F(n/N)) \ll (T/N)^{k+o(1)} N^{\ell +o(1)} \]

giving the claim.

Corollary 5.4 Exponent pairs closed and convex

The set of exponent pairs is closed and convex.

Proof ▶

Immediate from Lemma 5.3.

Proposition 5.5 Trivial exponent pairs

\((0,1)\) and \((1/2,1/2)\) are exponent pairs.

Implemented at exponent_pair.py as:
trivial_exp_pair

Proof ▶

Immediate from Lemma 5.3 and Lemma 4.4.

Heuristic 5.6 Exponent pairs conjecture

\((0,1/2)\) is an exponent pair. (Equivalently, by Corollary 5.4 and Proposition 5.5, every point in the triangle 1 is an exponent pair.)

Implemented at exponent_pair.py as:
exponent_pair_conjecture

Lemma 5.7

The exponent pair conjecture is equivalent to \(\beta (\alpha )=\alpha /2\) holding true for all \(0 \leq \alpha \leq 1\).

Proof ▶

Clear from Lemma 5.3 and Lemma 4.4.

Proposition 5.8 Van der Corput \(A\)-process

If \((k,\ell )\) is an exponent pair, then so is

\[ A(k,\ell ) := \left(\frac{k}{2k+2}, \frac{\ell }{2k+2} + \frac{1}{2}\right). \]

Recorded in literature.py as:
A_transform_hypothesis

Proof ▶

See [ 144 , Lemma 2.8 ] . It can also be deduced from Lemma 4.6 and Lemma 5.3.

Proposition 5.9 Van der Corput \(B\)-process

If \((k,\ell )\) is an exponent pair, then so is

\[ B(k,\ell ) := \left(\ell -\frac{1}{2}, k+\frac{1}{2}\right). \]

Recorded in literature.py as:
B_transform_hypothesis

Proof ▶

See [ 144 , Lemma 2.9 ] . Alternatively, this can be derived from Lemma 4.10 and Lemma 5.3.

5.1 Known exponent pairs

Proposition 5.10 Classical van der Corput exponent pairs

For any natural number \(k \geq 2\),

\[ A^{k-2} B(0,1) = \left( \frac{1}{2^k-2}, 1 - \frac{k-1}{2^k-2} \right) \]

is an exponent pair. In particular,

\[ \left(\frac{1}{2}, \frac{1}{2}\right), \left(\frac{1}{6}, \frac{2}{3}\right), \left(\frac{1}{14}, \frac{11}{14}\right) \]

are exponent pairs.

Proof ▶

Follows by induction from Proposition 5.8 and Proposition 5.5; alternatively, follows from (and is equivalent to) Corollary 4.8 and Lemma 5.3.

Derived in derived.py as:
van_der_corput_pair()

Corollary 5.11 Additional exponent pairs

The pairs

\[ \left(\frac{13}{31}, \frac{16}{31}\right), \left(\frac{4}{11},\frac{6}{11}\right), \left(\frac{2}{7},\frac{4}{7}\right), \left(\frac{5}{24},\frac{15}{24}\right), \left(\frac{4}{18},\frac{11}{18}\right) \]

are all exponent pairs.

Derived in derived.py as:
best_proof_of_exponent_pair(frac(13, 31), frac(16, 31))
best_proof_of_exponent_pair(frac(4, 11), frac(6, 11))
best_proof_of_exponent_pair(frac(2, 7), frac(4, 7))
best_proof_of_exponent_pair(frac(5, 24), frac(15, 24))
best_proof_of_exponent_pair(frac(4, 18), frac(11, 18))

Proof ▶

We have \((2/7,4/7) = BA(1/6,2/3)\), \((4/18,11/18) = BABA(1/6,2/3)\), and \((13/31, 16/31)=BAB^2A^2(1/6,2/3)\), so these cases follow from Propositions 5.10, 5.8, 5.9. Finally, \((4/11,6/11)\) is a convex combination of \((1/2,1/2)\) and \((2/7,4/7)\), and \((5/24, 15/24)\) is a convex combination of \((1/6,2/3)\) and \((4/18, 11/18)\), so these cases follow from Corollary 5.4.

Theorem 5.12 Exponent pairs on the line of symmetry

\((k,k+1/2)\) is an exponent pair for

\(k = 9/56\) [ 135 , Theorem 1 ] ;
\(k=89/560\) [ 293 , Theorem 6 ] ;
\(k=17/108\) [ 132 , p. 467 ] ;
\(k=89/570\) [ 128 , p. 40 ] ;
\(k=32/205\) [ 131 , Theorem 1 ] ;
\(k=13/84\) [ 23 , p. 206 ] .

Recorded in literature.py as:
add_literature_exponent_pairs()

Theorem 5.13 Exponent pairs from the Bombieri–Iwaniec method

The following pairs are exponent pairs:

\((\frac{2}{13}, \frac{35}{52})\) [ 136 ] ;
\((\frac{6299}{43860}, \frac{29507}{43860})\) [ 129 , Table 17.3 ] ;
\((\frac{771}{8116}, \frac{1499}{2029})\) [ 258 , p. 285 ] ;
\((\frac{21}{232}, \frac{173}{232})\) [ 258 , p. 286 ] ;
\((\frac{1959}{21656}, \frac{16135}{21656})\) [ 258 , p. 286 ] ;
\((\frac{516247}{6629696}, \frac{5080955}{6629696})\) [ 134 ] , [ 129 , Table 19.2 ] , [ 254 ] .

Recorded in literature.py as:
add_literature_exponent_pairs()

Theorem 5.14 Exponent pairs from derivative tests

\((k,1-mk)\) is an exponent pair when

\(k=\frac{1}{13}\) and \(m=3\) [ 254 , Theorem 1 ] ;
\(k = \frac{1}{204}\) and \(m=7\) [ 259 , p. 231 ] ;
\(k = \frac{1}{130}\) and \(m=8\) [ 249 , (1.1) ] ;
\(k = \frac{7}{2640}\) and \(m=8\) [ 259 , p. 231 ] ;
\(k = \frac{1}{716}\) and \(m=9\) [ 259 , p. 231 ] ;
\(k = \frac{1}{649}\) and \(m=9\) [ 253 ] ;
\(k = \frac{7}{4540}\) and \(m=9\) [ 249 , (1.2) ] ;
\(k = \frac{1}{615}\) and \(m=9\) [ 249 , (1.1) ] ;
\(k = \frac{1}{915}\) and \(m=10\) [ 250 , Th é or è me 2 ] .

Recorded in literature.py as:
add_literature_exponent_pairs()

Theorem 5.15 Huxley sequence

[ 129 , Table 17.3 ] For any integer \(m \geq 1\), the pair

\[ \left(\frac{169}{1424 \times 2^m - 338}, 1 - \frac{169}{1424 \times 2^m - 338} \frac{712m+1577}{712}\right) \]

is an exponent pair.

Recorded in literature.py as:
add_huxley_exponent_pairs(Constants.EXP_PAIR_TRUNCATION)

Theorem 5.16 1996 Heath–Brown sequence

[ 277 , (6.17.4) ] For any integer \(m \geq 3\), the pair

\[ \left(\frac{1}{25m^2 (m-2) \log m}, 1 - \frac{1}{25 m^2\log m}\right) \]

is an exponent pair.

(Currently not implemented in python due to the irrational exponents.)

Theorem 5.17 2017 Heath–Brown sequence

[ 113 , Theorem 2 ] For any integer \(m \geq 3\), the pair

\[ (p_m,q_m) := \left(\frac{2}{(m-1)^2(m+2)}, 1 - \frac{3m-2}{m(m-1)(m+2)}\right) \]

is an exponent pair.

Recorded in literature.py as:
add_heath_brown_exponent_pairs(Constants.EXP_PAIR_TRUNCATION)

Proof ▶

This follows from Theorem 4.23 and Lemma 5.3, after some computation.

Theorem 5.18 Sargos \(C\)-process

[ 259 , Theorem 5 ] If \((k,\ell )\) is an exponent pair, then so is

\[ \left(\frac{k}{12(1+4k)}, \frac{11(1+4k)+\ell }{12(1+4k)}\right). \]

Recorded in literature.py as:
C_transform_hypothesis

The following process is not quite a process to automatically transform one exponent pair to another, but it often achieves this in practice:

Theorem 5.19 Sargos \(D\)-process

[ 258 , Theorem 7.1 ] If \((k,\ell )\) is an exponent pair, then one has

\[ \beta (\alpha ) \leq \max \left( k_1 + \alpha (\ell _1-k_1), \frac{1}{12} + \frac{2}{3} \alpha \right) \]

for all \(0 \leq \alpha \leq 1\), where \((k_1,\ell _1) = D(k,\ell )\) is the pair

\[ D(k,\ell ) := \left(\frac{5k+\ell +2}{8(5k+3\ell +2)}, \frac{29k+21\ell +10}{8(5k+3\ell +2)}\right). \]

Recorded in literature.py as:
D_transform_hypothesis

Theorem 5.20

[ 279 , Lemma 1.1 ] The following are exponent pairs:

\begin{align*} (k_1,\ell _1) & := \left(\frac{4742}{38463}, \frac{35731}{51284}\right)\\ (k_2,\ell _2) & := \left(\frac{18}{199}, \frac{593}{796}\right)\\ (k_3,\ell _3) & := \left(\frac{2779}{38033}, \frac{58699}{76066}\right)\\ (k_4,\ell _4) & := \left(\frac{715}{10238}, \frac{7955}{10238}\right). \end{align*}

Recorded in literature.py as:
add_literature_exponent_pairs()

Proof ▶

For the pair \((18/199, 593/796)\), apply Theorem 5.19 with the pair \((13/84, 55/84)\) from Theorem 5.12 to conclude that

\[ \beta (\alpha ) \leq 18/199 + 521 \alpha / 796 \]

for all \(0 \leq \alpha \leq 1/2\), from which the claim follows from Lemma 5.3 (and Lemma 4.10). The remaining pairs come from Lemma 5.3 and the remaining components of Theorem 4.26.

Corollary 5.21 Set of exponent pairs

[ 279 , Theorem 1.3 ] Let \(H\) be the convex hull \((0,1)\), \((1/2,1/2)\), and of \((k_n,\ell _n)\) for \(n \in \mathbf{Z}\), where \((k_0,\ell _0) := 13/84\), \((k_n,\ell _n)\) for \(n=1,2,3,4\) is defined by Theorem 5.20, \((k_n,\ell _n) := A(k_{n-4},\ell _{n-4})\) for \(5 \leq n \leq 8\), \((k_n,\ell _n) := (p_n,q_n)\) for \(n {\gt} 9\) (with \((p_n,q_n)\) defined by Theorem 5.17), and \((k_{-n},\ell _{-n}) := B(k_n,\ell _n)\) for \(n \geq 0\). Then all elements of \(H\) are exponent pairs.

Indeed, as of [ 279 ] the set \(H\) represented all known exponent pairs, until Theorem 5.22 below.

Proof ▶

Clear from Corollary 5.4, Proposition 5.5, 5.20, and Theorem 5.17.

The following new exponent pairs were derived using this database:

Theorem 5.22 New exponent pairs

The following are exponent pairs:

\[ \left(\frac{89}{1282}, \frac{997}{1282}\right),\quad \left(\frac{652397}{9713986}, \frac{7599781}{9713986}\right),\quad \left(\frac{10769}{351096}, \frac{609317}{702192}\right),\quad \left(\frac{89}{3478}, \frac{15327}{17390}\right). \]

Derived in derived.py as:
prove_exponent_pair(frac(89,1282), frac(997,1282))
prove_exponent_pair(frac(652397,9713986), frac(7599781,9713986))
prove_exponent_pair(frac(10769,351096), frac(609317,702192))
prove_exponent_pair(frac(89,3478), frac(15327,17390))

Proof ▶

Using the bounds on \(\beta (\alpha )\) collected in Table 5.22, one may verify (after a tedious calculation) that for each of the claimed exponent pairs \((k, \ell )\) in the lemma statement, one has \(\beta (\alpha ) \le k + (\ell - k)\alpha \) for \(0 \le \alpha \le 1/2\). The result then follows from Lemma 4.10 and Lemma 5.3.

Table 5.1 Bounds on \(\beta (\alpha )\)

\(\beta (\alpha )\) bound

\(\alpha \) range

Reference

\(\dfrac {1}{20} + \dfrac {3}{4}\alpha \)

\(0\leq \alpha {\lt} \dfrac {1}{4}\)

Theorem 4.23 with \(k = 5\)

\(\dfrac {19}{20}\alpha \)

\(\dfrac {1}{4}\leq \alpha {\lt} \dfrac {890}{3277}\)

Theorem 4.23 with \(k = 5\)

\(\dfrac {89}{2706} + \dfrac {2243}{2706}\alpha \)

\(\dfrac {890}{3277}\leq \alpha {\lt} \dfrac {199}{716}\)

Table 4.1

\(\dfrac {1}{66} + \dfrac {235}{264}\alpha \)

\(\dfrac {120}{419}\leq \alpha {\lt} \dfrac {754}{2579}\)

Table 4.1

\(\dfrac {9}{217} + \dfrac {1389}{1736}\alpha \)

\(\dfrac {754}{2579}\leq \alpha {\lt} \dfrac {251324}{841245}\)

Exponent pair \((\dfrac {9}{217}, \dfrac {1461}{1736}) = AD(\dfrac {13}{84}, \dfrac {55}{84})\)

\(\dfrac {2371}{43205} + \dfrac {52209}{69128}\alpha \)

\(\dfrac {251324}{841245}\leq \alpha {\lt} \dfrac {861996}{2811205}\)

Exponent pair \((\dfrac {2371}{43205}, \dfrac {280013}{345640})\)

\(= A(\dfrac {4742}{38463}, \dfrac {35731}{51284})\) and Theorem 5.20

\(\dfrac {13}{146} + \dfrac {47}{73}\alpha \)

\(\dfrac {861996}{2811205}\leq \alpha {\lt} \dfrac {87}{275}\)

Table 4.1

\(\dfrac {11}{244} + \dfrac {191}{244}\alpha \)

\(\dfrac {87}{275}\leq \alpha {\lt} \dfrac {423}{1295}\)

Table 4.1

\(\dfrac {89}{1282} + \dfrac {454}{641}\alpha \)

\(\dfrac {423}{1295}\leq \alpha {\lt} \dfrac {227}{601}\)

Table 4.1

\(\dfrac {715}{10238} + \dfrac {3620}{5119}\alpha \)

\(\dfrac {227}{601}\leq \alpha {\lt} \dfrac {227}{601}\)

Exponent pair \((\dfrac {715}{10238}, \dfrac {7955}{10238})\)

in Theorem 5.20

\(\dfrac {29}{280} + \dfrac {173}{280}\alpha \)

\(\dfrac {227}{601}\leq \alpha {\lt} \dfrac {12}{31}\)

Table 4.1

\(\dfrac {1}{32} + \dfrac {103}{128}\alpha \)

\(\dfrac {12}{31}\leq \alpha {\lt} \dfrac {1508}{3825}\)

Table 4.1

\(\dfrac {18}{199} + \dfrac {521}{796}\alpha \)

\(\dfrac {1508}{3825}\leq \alpha {\lt} \dfrac {62831}{155153}\)

Exponent pair \((\dfrac {18}{199}, \dfrac {593}{796}) = D(\dfrac {13}{84}, \dfrac {55}{84})\)

\(\dfrac {569}{2800} + \dfrac {1053}{2800}\alpha \)

\(\dfrac {62831}{155153}\leq \alpha {\lt} \dfrac {143}{349}\)

Table 4.2

\(\dfrac {1}{12} + \dfrac {2}{3}\alpha \)

\(\dfrac {5}{12}\leq \alpha {\lt} \dfrac {3}{7}\)

Theorem 4.24

\(\dfrac {13}{84} + \dfrac {1}{2}\alpha \)

\(\dfrac {3}{7}\leq \alpha \leq \dfrac {1}{2}\)

Theorem 4.24

Furthermore, more exponent pairs can be derived upon incorporating Lemma 4.6.

Theorem 5.23 Cushing (2025) exponent pairs

The following are exponent pairs:

\[ \left(\frac{311}{4822}, \frac{3799}{4822}\right),\quad \left(\frac{80219}{1298878}, \frac{515638}{649439}\right). \]

Implemented at examples.py as:
beta_bound_examples2()

In summary, the current set of known exponent pairs is the convex hull with vertices \((0, 1)\), \((1/2, 1/2)\) and the points \((k_n, \ell _n)\) for \(n \in \mathbf{Z}\) that are recorded in Table 5.23.

Table 5.2 Vertices of the convex hull of known exponent pairs.

\(n\)

\((k_n, \ell _n)\)

Reference

\(\left(\dfrac {13}{84}, \dfrac {55}{84}\right)\)

[ 23 , p. 307 ]

\(\left(\dfrac {4742}{38463}, \dfrac {35731}{51284}\right)\)

Theorem 5.20

\(\left(\dfrac {18}{199}, \dfrac {593}{796}\right)\)

Theorem 5.20

\(\left(\dfrac {2779}{38033}, \dfrac {58699}{76066}\right)\)

Theorem 5.20